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On the slopes of the U5operator acting on overconvergent modular forms Tome 20, no1 (2008), p. 165-182.

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de Bordeaux 20 (2008), 165-182

On the slopes of the U

5

operator acting on

overconvergent modular forms

par L. J. P KILFORD

Résumé. Nous démontrons que les pentes de l’opérateur U5

agis-sant sur 5-adique formes modulaires surconvergentes de poids k avec caractère de Dirichlet primitif χ de conducteur 25 sont

 1 4·  8i 5  : i ∈ N  ou  1 4·  8i + 4 5  : i ∈ N  .

Nous prouvons aussi que l’espace de forms parabolique de poids k et caractère χ a une base des formes propres pour les opérateurs de Hecke Tp et U5 définie sur Q5(4

√ 5,√3).

Abstract. We show that the slopes of the U5operator acting on

5-adic overconvergent modular forms of weight k with primitive Dirichlet character χ of conductor 25 are given by either

 1 4 ·  8i 5  : i ∈ N  or  1 4 ·  8i + 4 5  : i ∈ N  , depending on k and χ.

We also prove that the space of classical cusp forms of weight k and character χ has a basis of eigenforms for the Hecke opera-tors Tp and U5which is defined over Q5(

4 √

5,√3).

1. Introduction

We first define the slope of a (normalized) cuspidal eigenform.

Definition 1. Let f be a normalized cuspidal modular eigenform with Fourier expansion at ∞ given by P∞

n=1anqn. The slope of f is defined

to be the 5-valuation of a5 viewed as an element of C5; we normalize the 5-valuation of 5 to be 1.

As a consequence of the main result of this paper, we will prove the following theorem about the slopes of classical modular forms. Let us set up some notation.

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The cyclotomic polynomial Φ20(x) factors over Q5 into two factors, f1 and f2, such that

f1 ≡ x4+ 2x3+ 4x2+ 3x + 1 mod 5, (1)

f2 ≡ x4+ 3x3+ 4x2+ 2x + 1 mod 5. (2)

Let χ be an odd primitive Dirichlet character of conductor 25 and let τ be an odd primitive Dirichlet character of conductor 5.

Let k be a positive integer. We fix an embedding of the field of definition of χ into Q5(4

5,√3). Then we have the following theorem which tells us what the slopes of the eigenforms are:

Theorem 2. Using the preceding notation, the slopes of the U5 operator acting on Sk(Γ0(25), χτk−1) are 1 4 · b 8i 5c : i ∈ N  if χ(6) is a root of f1, 1 4 · b 8i + 4 5 c : i ∈ N  if χ(6) is a root of f2.

Because all of the slopes are distinct, we can also prove a result about the field of definition of a basis of eigenforms for this space.

Corollary 3. Let k be a positive integer and let χ be a primitive Dirichlet character of conductor 25. Then if f is a normalized modular eigenform of weight k and character χ, its Fourier expansion is defined over Q5(4

√ 5,√3). This gives some evidence towards Questions 4.3 and 4.4 of Buzzard [2]; these concern conjectural bounds on the degree of the field of definition of certain modular forms over Qp.

2. Previous work, and new directions

This paper uses methods introduced by Emerton in his PhD thesis [9], which deals with the action of the U2 operator. It also uses methods devel-oped by Smithline in his thesis [16] and applied by him in [17], which were then also used in the author’s paper [11] and in the paper of the author with Buzzard [4].

In [17], the following theorem is proved about 3-adic modular forms:

Theorem 4 (Smithline [17], Theorem 4.3). We order the slopes of U3 by

size, beginning with the smallest.

The sum of the first x nonzero slopes of the U3 operator acting on 3-adic

overconvergent modular forms of weight 0 is at least 3x(x − 1)/2 + 2x, and is exactly that if x is of the form (3j − 1)/2 for some j.

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His thesis also shows that the sum of the first x slopes of the U5 operator acting on 5-adic overconvergent modular forms of weight 0 is at least x2.

In Buzzard-Kilford [4], the following theorem was proved about the 2-adic slopes of U2 acting on certain spaces of modular forms.

Theorem 5 (Buzzard-Kilford [4], Theorem B). Let k be an integer and let χ be a character of conductor 2n such that χ(−1) = (−1)k.

If |5k· χ(5) − 1|2 > 1/8, then the slopes of the overconvergent cuspidal

modular forms of weight k and character χ are {t, 2t, 3t, . . .}, where t =

v(5k· χ(5) − 1), and each slope occurs with multiplicity 1.

This paper will prove a 5-adic analogue of these results for the specific level Γ0(25). However, the slopes are no longer in one arithmetic progres-sion, as they are in the previously studied cases. Instead, there are five arithmetic progressions which interlace together; these all have a common difference between terms (which is 2). (One could, of course, view the arith-metic progression 1, 2, 3, . . . as being made up of the two aritharith-metic pro-gressions 1, 3, 5, . . . and 2, 4, 6 . . . but this point of view is only reasonable after one has considered the action of U5 on forms of level 25, where the slopes form several arithmetic progressions).

In Buzzard-Calegari [3], the following theorem is proved, using similar techniques to those in [16], [11] and [4]:

Theorem 6. The slopes of the U2 operator acting on 2-adic overconvergent modular forms of weight 0 are

 1 + 2v2 (3n)! n!  : n ∈ N  ,

where v2 is the normalized 2-adic valuation.

We see that the slopes at weight 0 do not fall into the arithmetic pro-gressions found in [11] and [4]; we note that it appears that the behaviour of the slopes at the boundary of weight-space, which we consider in this paper, is rather different to the behaviour of the slopes at the centre of weight-space considered in [3].

In addition to the fact that the slopes do not lie in one arithmetic pro-gression, there are some other interesting new features which appear when we consider 5-adic overconvergent modular forms that do not appear for 2-adic or 3-adic overconvergent modular forms.

Firstly, there is a computational issue. In previous work (such as [4] or [11]), the computations in magma [1] could be carried out either over the rational numbers or over the field Qp. However, for p = 5 the calculation must be carried out over Q5(4

√ 5,√3).

Secondly, and more importantly, there are now two different possibilities for the slopes, which depend on which character is chosen. These two pos-sibilities correspond to the two factors of the cyclotomic polynomial Φ20(x)

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over Q5. This is a departure from the situation in [4] and [11], where the slopes are independent of choice of character.

Thirdly, part of the complexity in the calculations in [11] was the fact that (in the notation of that paper) the modular function U2(z2i+1) was identically zero. This meant that the first “matrix of the U2 operator” that was defined had identically zero determinant, which meant that some algebra had to be done to get a matrix to which Theorem 13 could be applied to. In the current note, this does not happen because the matrix M of the U5 operator does not have any identically zero columns.

Finally, the strategy of Section 4 was chosen because the modular func-tions involved were unusually simple, thus making the calculafunc-tions more tractable (the corresponding functions for (say) weight 2 were much less pleasant).

3. Defining 5-adic overconvergent modular forms

We now present the definition of the 5-adic overconvergent modular forms, first by defining overconvergent modular forms of weight 0, and then by defining forms with weight and character in terms of the weight 0 forms and a suitable Eisenstein series.

This section follows Section 3 of [11] in its layout and direction; more details on the specific steps can be found there.

Following Katz [10], section 2.1, we recall that, for C an elliptic curve over an F5-algebra R, there is a mod 5 modular form A(C) called the Hasse

invariant, which has q-expansion over F5 equal to 1.

We consider the Eisenstein series of weight 4 and tame level 1 defined over Z, with q-expansion

E4(q) := 1 + 240 ∞ X n=1   X 0<d|n d3  · qn.

We see that E4is a lifting of A(C) to characteristic 0, as the reduction of E4 to characteristic 5 has the same q-expansion as A(C), and therefore E4 mod 5 and A(C) are both modular forms of level 1 and weight 4 defined over F5, with the same q-expansion. Note also that if C is an elliptic curve defined over Z5 then the valuation v5(E4(C)) can be shown to be well-defined.

It is interesting to note that one can use the same Eisenstein series, E4, in this part of the definition for 2-adic, 3-adic and 5-adic overconvergent modular forms (as a lifting of the 4th, 2nd and 1st power of the Hasse invariant, respectively).

We now let m be a positive integer, and we recall that the modu-lar curve X0(pm) is defined to be the moduli space that parametrizes

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pairs (C, P ), where C is an elliptic curve and P is a subgroup of C of order pm.

Using arguments exactly similar to those in [11], we define the affinoid subdomain Z0(5m) of X0(5m) to be the connected component containing the cusp ∞ of the set of points t = (C, P ) in X0(5m) which have v5(E4(t)) = 0; we note that v5(E4(t)) = 0 means either that the point t corresponds to an ordinary elliptic curve or that t is a cusp.

We now define strict affinoid neighbourhoods of Z0(5m).

Definition 7 (Coleman [7], Section B2). We think of X0(5m) as a rigid

space over Q5, and we let t ∈ X0(5m)(Q5) be a point, corresponding either

to an elliptic curve defined over a finite extension of Q5, or to a cusp. Let w

be a rational number, such that 0 < w < 52−m/6.

We define Z0(5m)(w) to be the connected component of the affinoid {t ∈ X0(5m) : v5(E4(t)) ≤ w}

which contains the cusp ∞.

Given this definition, we can now define 5-adic overconvergent modular forms.

Definition 8 (Coleman, [6], page 397). Let w be a rational number, such that 0 < w < 52−m/6. Let O be the structure sheaf of Z0(5m)(w). We

call sections of O on Z0(5m)(w) w-overconvergent 5-adic modular forms of weight 0 and level Γ0(5m). If a section f of O is a w-overconvergent

modular form, then we say that f is an overconvergent 5-adic modular

form.

Let K be a complete subfield of C5, and define Z0(5m)(w)/K to be the

affinoid over K induced from Z0(5m)(w) by base change from Q5. The space

M0(5m, w; K) := O(Z0(5m)(w)/K)

of w-overconvergent modular forms of weight 0 and level Γ0(5m) is a K-Banach space.

We now let χ be a primitive Dirichlet character of conductor 5m and let k be an integer such that χ(−1) = (−1)k. Let Ek,χbe the normalized Eisenstein series of weight k and character χ with nonzero constant term. The space of w-overconvergent 5-adic modular forms of weight k and character χ is given by

Mk,χ(5m, w; K) := Ek,χ∗ · M0(5m, w; K).

This is a Banach space over K.

There are Hecke operators U5 and Tp (where p - 5) acting on the space of modular forms Mk,χ(5m, w; K); these are defined on the q-expansions of the overconvergent modular forms in exactly the same way as they are

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defined on the q-expansions of classical modular forms. One defines Tnfor n a natural number in the usual way.

Using results of Coleman, we have the following theorem about the inde-pendence of the characteristic power series of U5 acting on Mk,χ(5m, w; K):

Theorem 9 (Coleman [7], Theorem B3.2). Let w be a real number such that 0 < w < min(52−m/6, 1/6), let k be an integer and let χ be a character

such that χ(−1) = (−1)k.

The characteristic polynomial of U5 acting on w-overconvergent 5-adic

modular forms of weight k and character χ is independent of the choice of w.

We will now rewrite the definition of Z0(25)(w) in terms of a carefully chosen modular function of level 25, in order to prove the following theorem:

Theorem 10. Let w0 = 1/12. The space of w0-overconvergent modular

forms of weight 0 and level 25, with coefficients in Q5(√45), is a Tate algebra

in one variable over Q5(4 √

5).

Proof. We have given a valuation on the points t of the rigid space X0(5m), based on the lifting of the Hasse invariant by the Eisenstein series E4. We recall that the modular j-invariant is defined to be j := E43/∆. Therefore, we see that, if the elliptic curve corresponding to t has good reduction, then ∆(t) has valuation 0, and therefore that

v5(t) = v5(E4(t)) = 1

3 · v5((E4)(t) 3) = 1

3 · v5(j(t)).

We now recall that the modular curve X0(25) has genus 0. This means that there is a modular function t25 which is a uniformizer on X0(25):

t25:= η(q) η(q25),

where η is the Dedekind η-function. We could write t25as a rational function in j directly, but as the resulting rational function is very complicated, we will instead also work with the uniformizer t5 of X0(5), defined as

t5 :=

η(q)

η(q5)

6

By explicit calculation, one can verify the following identities of modular functions: (3) j = (t 2 5+ 250t5+ 3125)3 t55 and t5= t525 t425+ 5t325+ 15t225+ 25t25+ 25. We note also that

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this follows because the q-expansion of all of these functions begins q−1+· · · . Let D be an element of C5 such that 0 < v5(D) < 1/2. We see that, if 0 ≤ v5(j) ≤ D, then v5(j) = v5(t5) (the valuation of t25+ 250t5 + 3125 is given by the valuation of t25) and that v5(t5) = v5(t25) (because the val-uation of t425+ 5t325+ 15t225+ 25t25+ 25 is given by the valuation of t425). Therefore, for this range of D we have that v5(j) = v5(t25). Then, be-cause t5(∞) = t25(∞) = ∞, this means that the connected components of Z0(5) and Z0(25) which contain the cusp ∞ of sufficiently small radius are of the form v5(t5) < D1and v5(t25) < D2, for some rational numbers D1 and D2 with valuations between 0 and 1/2.

By considering the Newton polygons of the numerators and denominators of the rational functions in (3), we see that if v5(t25) < 1/2, then v5(t25) = v5(t5) = v5(j). This means that we have shown that

Z0(25)(w) = {x ∈ X0(25) : v5(t25(x)) ≤ 3w} , for 0 < w < 1/6. Now, we choose w = 1/12, and therefore we obtain

Z0(25)(1/12) = {x ∈ X0(25) : v5(t25(x)) ≤ 1/4} .

Let us define W := √45/t25. We can rewrite the definition of Z0(25)(1/12) again in terms of W to get

Z0(25)(1/12) = {x ∈ X0(25) : v5(W (x)) ≥ 0} .

Finally, we recall that the rigid functions on the closed disc over Q5with centre 0 and radius 1 are defined to be power series of the form

X

n∈N

anzn : an∈ Q5, an→ 0.

Therefore, the space of 1/12-overconvergent modular forms of level Γ0(25) and weight 0 is

Q5(

4

√ 5)hW i,

which is what we wanted to show. 

We have written down this space of overconvergent modular forms as an explicit Banach space. This means that we can write down its Banach

basis: the set 

W, W2, W3, . . .

forms a Banach basis for the overconver-gent modular forms of weight 0 and level Γ0(25). This Banach basis is composed of weight 0 modular functions — we want to be able to con-sider the action of the U5 operator on overconvergent modular forms with non-zero weight k and character χ (here, as elsewhere in this note, χ has conductor 25 and χ(−1) = (−1)k). Using an observation from the work of Coleman [7], we will be able to move between weight 0 and weight k and character χ via multiplication by a suitable quotient of modular forms.

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Let F be an overconvergent modular form of weight k and character χ which has nonzero constant term, and let z be an overconvergent modu-lar function of weight 0. In particumodu-lar, we note that F may have negative weight. From the discussion in Coleman [7, page 450] we see that the pull-back ˜U5 of the U5 operator acting on overconvergent modular forms of weight k and character χ to weight 0 is 1/F · U5(z · F ).

Now by equation 3.3 of [8] we have that (4) U5(z · V (F )) = U5(z) · F.

We therefore consider the modular form V (F ), which has nonzero constant term because F does, and substitute V (F ) into the formula for the pullback of U5 to weight zero, to obtain

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V (F ) · U5(z · V (F )) = F

V (F )· U5(z).

We can also use (4) to see that the pullback of U5 acting on overconvergent modular forms of weight k and character χ is

(6) 1/F · U5(z · F ) = U5  z · F V (F )  .

We now use equations (5) and (6) to define the twisted U5 operatorU .e

Definition 11 (The twisted U5 operator). Let k be an integer and let χ be

an odd character of conductor 25. Let τ be an odd character of conductor 5, and let W be the overconvergent modular form √4

5/t25 of level Γ0(25) and

weight 0 that we defined above.

We defineU to be the “weight k” twisted U5e operator acting on the basis

{Wi· E∗ 1,χ/V (E1,χ∗ )} by the formula: (7) U (We i) := U5 Wi· E1,χ V (E1,χ) ! · E1,τ V (E1,τ) !k−1 .

We note here that the operatorU acts on overconvergent modular formse

of weight 0; we consider it because it has the same characteristic power series as the standard U5 operator acting on overconvergent modular forms with nonzero weight and character.

We can now consider the action of the operator U on these spaces ofe

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Definition 12 (The matrix of the twisted U5 operator). Let k be an integer

and let χ be an odd character of conductor 25. Let τ be a character of conductor 5 such that χ(−1) = (−1)k. We let W be the modular function given in Definition 11.

Let M = (mi,j) be the infinite compact matrix of the twisted U5 operator

acting on overconvergent modular forms of weight 1+k and character χ·τk, where mi,j is defined to be the coefficient of Wi in the W -expansion of the

operator defined in equation (7).

Here we will make the observation that the entries of our matrix M depend on χ and τ .

We know that U5 is a compact operator, so we can show that the trace, determinant and characteristic power series of M are all well-defined. The following result follows directly from the general result given as Proposi-tion 7 of [15], on the characteristic power series of compact operators (Serre uses the older term “completely continuous” for what we are calling “com-pact”).

Theorem 13. (1) Let Mn be an n × n matrix defined over a finite extension of Qp. Let det(1 − tMn) = Pni=0citi. Let Mm be the

matrix formed by the first m rows and columns of Mn.

Let s(i) be the formula for the ith slope; in our specific case, this will mean that either

s(i) = 1 4 · 8i 5  or s(i) = 1 4 · 8i + 4 5  .

Assume that there exists a constant r ∈ Q× such that

(a) For all positive integers m such that 1 ≤ m ≤ n, the valuation

of det(Mm) is r ·Pmi=1s(i).

(b) The valuation of elements in column j is at least r · s(i).

Then we have that, for all positive integers m such that 1 ≤ m ≤

n, v2(cm) = r ·Pm

i=1s(i).

(2) Let M∞ be a compact infinite matrix (that is, the matrix of a com-pact operator). If Mm is a series of finite matrices which tend

to M, then the finite characteristic power series det(1 − tMm)

converge coefficientwise to det(1 − tM∞), as m → ∞.

We now quote a result of Coleman that tells us that overconvergent modular forms of small slope are in fact classical modular forms:

Theorem 14 (Coleman [6], Theorem 1.1). Let k be a non-negative integer and let p be a prime. Every p-adic overconvergent modular eigenform of weight k with slope strictly less than k − 1 is a classical modular form.

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We now state the main theorem of this paper, which tells us exactly what the slopes of the U5 operator acting on slopes of modular forms of level 25 are.

Theorem 15. We recall and use the notation of Theorem 2.

Let χ be an odd primitive Dirichlet character of conductor 25 and let τ be an odd primitive Dirichlet character of conductor 5.

Let k be a positive integer. We fix an embedding of the field of definition of χ into Q5(4

5), and recall the notation of f1 and f2 from Theorem 2.

The slopes of overconvergent modular forms of weight k and charac-ter χτk−1 are given by

1 4 · b 8i 5c : i ∈ N  if χ(6) is a root of f1, 1 4 · b 8i + 4 5 c : i ∈ N  if χ(6) is a root of f2.

We can prove Theorem 2, assuming Theorem 15, by recalling the follow-ing theorem from Cohen-Oesterlé:

Theorem 16 (Cohen-Oesterlé [5], Théorème 1). Let χ be a primitive Dirichlet character of conductor 25 and let k be a positive integer greater than 1. The following formula holds:

d(k, χ) := dim Sk(Γ0(25), χ) = 5k − 7

2 + ε · (χ(8) + χ(17)),

where ε is 0 for odd k, −1/4 if k ≡ 2 mod 4, and 1/4 if k ≡ 0 mod 4. Proof of Theorem 2. The classical theorem will follow, because when we

substitute d(k, χ) into the formula s(i) for the ith slope, we see that the maximum value of s(d(k, χ)) is k − 1. We now apply either Theorem 14 or a standard argument shows that slopes greater than k − 1 cannot be classical (see [4], the proof of the Corollary to Theorem B, which references the proof of Theorem 4.6.17(1) of [13]), so therefore as there are at most k − 1 slopes which are smaller than or equal to k − 1, we see that all of these small slopes are the slopes of classical eigenforms.  Finally, we show that there is a basis of eigenforms which is defined over K := Q5(4

5,√3), to prove Corollary 3.

Proof of Corollary 3. Let χ be a primitive Dirichlet character of

conduc-tor 25 and let k be a positive integer such that χ(−1) = (−1)k. We recall a useful fact stated on page 21 of [14], that if f (q) =P∞

n=1anqn is the Fourier expansion of a nonzero normalized classical modular cuspidal

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eigenform of weight k and character χ, and σ is an element of Gal(K/K), then σ(f )(q) := ∞ X n=1 σ(an)qn

is also the Fourier expansion of a normalized classical modular cuspidal eigenform of weight k and character χ; we see that because χ takes values in K, it is invariant under the action of σ.

Now, we see that the 5-valuation of a5 is the same as the 5-valuation of σ(a5), because the characteristic polynomial of a5 is invariant under the action of σ. Therefore, we see that σ(f ) is an eigenform of the same weight and character as f with the same slope as f , because there is only one eigenform of any given slope for any given χ and k. This means that σ(f ) = f for every choice of σ, so therefore σ(an) = an for every σ, which means

that an∈ K for every n, as required. 

4. The technical part; proof of Theorem 15

As the actual proof of Theorem 15 is somewhat technical, we will first outline a plan to show how the proof works.

Plan for the proof of Theorem 15. In this section, we will show that

we can apply Theorem 13, which will prove Theorem 15. First we fix an ar-bitrary positive integer n, an integer k and a primitive Dirichlet character χ of conductor 25 such that χ(−1) = −1.

We will begin with the matrix Mn; the matrix formed by the first n rows and n columns of M , the matrix of the twisted U5operator acting on forms of weight 1 and character χ defined in Definition 11. The proof will then proceed in the following way:

(1) Define the matrix D(β(i)) to be the diagonal matrix with β(j) in

the jthrow and the jthcolumn. We define the matrix On:= D(π−2j)· Mn· D(π2j).

(2) We then show that the valuation of elements in the jth column of On

are s(j); this verifies condition (b) of Theorem 13, with r = s(j).

(3) We finally show that On has determinant of valuation Pn

i=1s(i), by

considering the matrix Pn:= D(π−4s(j))·On. We will see that Pnhas

determinant of valuation 0, which implies that the valuation of the determinant of On is the valuation of the determinant of D(π4s(j)),

which is Pn

i=1s(i). This will verify condition (a) of Theorem 13,

with determinant of valuation Pn

i=1s(i).

(4) Finally, we will show that, after multiplication by the multiplier (as

defined in Definition 12)

E1,τ V (E1,τ)

!k

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the matrix of the twisted U5operator acting on forms of weight 1+k

and character χ · τk−1 still satisfies properties (a) and (b) of Theo-rem 13.

At each step of this plan, we must show that the characteristic polynomial of the new matrix defined is the same as that of Mn. In the last step, we will show that Pnhas unit determinant by reducing it modulo a prime ideal above 5 and showing that this reduction has determinant 1. This means that we must prove that Pn has coefficients which are integers in Q5(√45).

Proof of Theorem 15. In this section, we will use the modular function T

instead of W ; we define T as follows: T := 1

t25.

This will make it easier to perform the calculations.

We will adapt a method used by Smithline in [16] and [17] to find the matrix Mn of the “weight 1” operator U , which will use the crucial facte

that this operator is what Smithline calls “a compact operator with rational generation”. This means thatU (Te i) satisfies a recurrence relation in terms

of otherU (Te j), for j = i − 5, . . . , i − 1.

Firstly, we can show thatU (Te i) is a polynomial in T of degree 5i, for 1 ≤

i ≤ 5. We prove this by explicit computation; we can see that these are polynomials by studying the zeroes and poles of theU (Te i), or by an explicit

computation of sufficiently many Fourier coefficients of both sides. This directly generalizes the work of Smithline on the weight 0 operator Up.

Secondly, we will show that a recurrence relation exists, and then find an easy to calculate form in terms of the U (Te j) and T . This will allow us

to compute the determinant of the matrix of U acting on the basis {Te i}

and therefore to compute the slopes of the characteristic power series ofU .e

We will just give the valuations of the coefficients of these, as the actual coefficients are elements of Q5(√45) and thus take up a lot of space. If we have chosen χ such that χ(6) is a root of (1), then these are the valuations of the coefficients of T in the T -expansions of theU (Te i) for 1 ≤ i ≤ 5:

e U (T ) : h14,54, 2, 3, 4i e U (T2) : h14,34,47,52, 4,174,214, 6, 7, 8i e U (T3) : h0,34, 1, 2, 3,174 ,194,234 ,132, 8,334,374, 10, 11, 12i e U (T4) : h0,12, 1,23, 3,134 ,174, 5, 6, 7,172,354 ,394,212 , 12,494,534 , 14, 15, 16i e U (T5) : h0, 1, 1, 2, 2,72,154,194 ,112, 7,294,334 ,374,414 , 11,494,514,554,292 , 16,654 ,694, 18, 19, 20i.

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On the other hand, if we have chosen χ(6) to be a root of (2), then these are the valuations of the coefficients of T in the T -expansions of theU (Te i):

e U (T ) : h12,32,94,134, 4i e U (T2) : h12, 1, 2,114 , 4,29,112,254 ,294, 8i e U (T3) : h14, 1,54,94, 3, 5, 5, 6,274, 8,172,192,414,454 , 12i e U (T4) : h14,34,45,74, 3,72,92,214,254 , 7,172 , 9, 10,434, 12,252 ,272,574 ,614, 16i e U (T5) : [0, 1, 1, 2, 2,72, 4, 5,234, 7,152,172 ,192,212, 11, 25 2, 13, 14, 59 4, 16, 33 2, 35 2 , 73 4, 77 4 , 20]

(We note here that these are the valuations of the elements of the Ti-coefficients of theU (Te j) viewed as elements of Q5(

4

5,√3)).

The T -expansion of U (T ) is not dependent on the choice of valuations; it is

U (T ) = 5T + 25T2+ 75T3+ 125T4+ 125T5,

which is [1, 2, 2, 3, 3] in the valuation scheme we have used for the U (Te j)

above.

There is an explicit recurrence relation which theU (Te i) satisfy; we have

that e U (Ti+5) = 1 5· U5(T ) · (25U (Te i+4) + 25 e U (Ti+3) (8)

+ 15U (Te i+2) + 5U (Te i+1) +U (Te i)), for i ≥ 0.

This, together with the explicit values for U (Te i) given above, will allow

us to determine all of the U (Te i). We see by induction on i that all of

the U (Te i) are polynomials in T of degree 5i; this, together with the fact

that the relation involves powers of 5, also allow us to show that the matrix of the operator U is compact.e

We also note that, because 5 divides the coefficients of U5(T ), thatU (Te i)

is an integral polynomial in T .

We prove this recurrence relation by first showing that the same recur-rence relation as in (8) holds for U5(Ti). Following the lucid account given in [12], Section 2, we recall that

U5(Tj)(z) = 1 5 · X t∈Z/5Z Tj z + t 5  .

If we define ek to be the kth symmetric polynomial in the {T (z+t5 ) : t ∈ Z/5Z}, then we see that we have the following recurrence relation:

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Now, we can show that e1 = 25U5(T ), e2 = −25U5(T ), e3 = 15U5(T ), e4 = −5U5(T ) and e5 = U5(T ) by explicit computations of the symmetric polynomials in the T (z+t5 ). For instance, we see that e1 = 25U5(T ) because that is how U5(T ) is defined. For the others, we can compute these by changing the variable z to 5z and then using Magma to compute the ei to a sufficient precision to show that they are the correct multiples of U5(T ). It is well-known that every modular function of weight 0 for Γ0(25) must be a rational function of T , which is a uniformizer of the genus 0 curve X0(25). Because the ek have no poles on X0(p) except possibly at the cusp 0, they must actually be polynomials in T , and we can compute these polynomials effectively.

We now derive the recurrence relation for U (Te j) from that for U5(Tj).

By definition, we have that

e U (Tj) = U5(Tj · E1,χ/V1,χ)(z) = 1 5 X t∈Z/5Z Tj z + t 5  · (E1,χ/V1,χ) z + t 5  . (9)

Now, the functions Ti((z + t)/5), for t ∈ {0, 1, 2, 3, 4}, are the general solutions to the recurrence relation, because they are the roots of the poly-nomial associated to the recurrence. Therefore, if F is fixed, both U (Ti) and U (Ti· F ) satisfy the same recurrence relation, so in particular we see that theU (Te i) do satisfy a recurrence relation.

We note in passing that similar calculations to those in this section can be performed for modular functions on X0(4) and X0(9), and similar recur-rence relations derived, involving a common multiple of a suitable equiva-lent of U5(T ) as in (8).

Using this recurrence relation, we could derive a generating function forU , in the style of [3] or [17], but this would be hard to display and note

very illuminating. We will instead use the recurrence relation in the next Section to show that the valuations of the coefficients of the ith column of the matrix On are at least s(i), that the valuation of the (i, i)th entry of Onis exactly s(i), and that the valuation of elements below the diagonal in the ith column is strictly greater than s(i). This will show that the determinant of On has valuationPnj=1s(j), which will be enough to prove our theorem.

4.1. Finding the valuation of elements of On. In this section we will define X := π2T and U (Xi) := π2i·U (Xe i) for clarity of notation.

We check that the X-expansion of U (Xr) has integral coefficients, for r = 1, . . . , 5, that the valuation of the coefficient of Xr in U (Xr) is s(i), and that the valuation of the coefficient of Xj in U (Xr) is greater than s(i) for j > i. We call these statements Ir for short.

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We will now prove by induction that Ir holds for every positive integer r. Firstly, we note that the base cases are satisfied; we can check that the valuations of the T -coefficients of theU (Te r) given above, when translated

into the language of Xs, satisfy Ir for r = 1, . . . , 5.

We suppose that Is holds for every positive integer s < r. We now use the recurrence relation given in (8) to show that

U (Xr) = (X + π2X2+ 3X3+ π2X4+ X5) · (25 · U (Xr−1) + 25π2U (Xr−2) + 15π4U (Xr−3) + 5π6U (Xr−4) + π8· U (Xr−5)).

We see that the elements of strictly lowest valuation in this sum are those from the term U := π8 · U (Xr−5) · (X + π2X2+ 3X3+ π2X4 + X5), so the valuations of U (Xr) are determined by this product. We also note that the valuation of the coefficient of Xr in U is exactly s(r − 5) + 2, that the valuations of the coefficients of Xs for s > r in U are greater than s(r − 5) + 2, and that the valuation of every coefficient of U (Xr) is at least s(r − 5) + 2 (by using Ir−5). Therefore we have proved Ir by induction, because s(r) = s(r − 5) + 2.

We now postmultiply the matrix On by D(π−4s(i)) and define this prod-uct to be Pn. The effect on the U (Xi) is to multiply them by π−4s(i). Be-cause the elements of Pnare given by the coefficients of these X-expansions, we see that Pn has integral coefficients, that the coefficients on the diag-onal are units, and that the coefficients below the diagdiag-onal have strictly positive valuation. We can therefore reduce Pn modulo π and take its de-terminant; we see that Pn must have unit determinant by considering the main diagonal.

This means that the determinant of Onand also the determinant of Mn both have valuationPn

i=1s(i). This means that Mnsatisfies condition (a) of Theorem 13 and therefore that we can apply this theorem to the matrix M to show that the slopes of the U5 operator are given by s(i).

4.2. Generalizing all this to other weights. We note that this part

of the proof has shown that the matrices Mn of the twisted U5 operator acting on overconvergent modular forms of weight 1 have determinants with valuation Pm

i=1s(i), and that the valuations of elements in the jth column are at least s(i). We also note that we have expressed On in the form On = Pn· D, where D is a diagonal matrix and P is a matrix which is upper triangular and invertible modulo π.

We will now prove that the matrix of the twisted U5 operator acting on weights of the form 1 + k, where k is an integer, also satisfies these two properties; this will be enough to prove Theorem 15.

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We note that the following identity of modular functions holds: (10) E1,τ V (E1,τ) = 1 − 5(T + (2 + 2I)T 2) 1 + (2 + I)T + (2 + I)T2,

where τ is an odd primitive Dirichlet character of conductor 5, and I is a suitably chosen square root of −1.

From this, we see that the following Lemma holds:

Lemma 17. Let O be the ring of integers of Q5(π). Then

(11) E1,τ

V (E1,τ)

∈ 1 + XO[[X]].

Proof of Lemma 17. We recall that X = π2T , and check that when we substitute this into (10) that the coefficients of the Xj remain integral.  We now change weights from weight 1 to weight 1 + k; this means that we change the basis on which the Ue5 operator acts by multiplication by

the power series (E1,τ/V (E1,τ))k. We see that this change of basis has the property that Xi is sent to a power series in X whose exponents are at least i, and that the coefficient of Xi in this new power series is 1. Also, by (11), we see that these power series have integral coefficients. We call the matrix which gives this change of basis S.

From this, we see that this matrix S has the following properties: it has coefficients in the ring of integers O, the diagonal entries of S are all 1, and it is lower triangular modulo π.

The change of basis by S has the effect of replacing On by Pn· D · S = Pn· (D · S · D−1) · D.

We will now show that Pn· D · S · D−1 is invertible and upper triangular modulo π. The action of conjugation by D on S is to divide elements above the diagonal by π4s(i)and to multiply elements below the diagonal by π4s(i); given the list of properties of S above, we see that this means that D·S ·D−1 must be the identity modulo π, which means that if we premultiply by Pn then the product will be upper triangular and invertible modulo π.

This means that we are in the same situation as before; the determinant of the n × n truncation of the matrix of the operatorU5e acting on

overcon-vergent modular forms of weight 1 + k has valuationPn

i=1s(i). This means that the matrix satisfies condition (a) of Theorem 13 and therefore that we can apply this theorem to show that the slopes of the U5 operator are given by s(i), which is what we wanted to prove. 

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5. Acknowledgements

A preliminary version of this article was written during my stay at the Max-Planck-Institut in Bonn, Germany in the summer of 2004. The author would like to thank the Institut for its hospitality.

The results of this paper were mentioned in a short note circulated at the Banff International Research Station workshop “p-adic variation of mo-tives”, which was held in December 2003. The author would also like to thank the BIRS for its hospitality.

Some of the computer calculations were executed by the computer al-gebra package Magma [1] running on the machine crackpipe, which was bought by Kevin Buzzard with a grant from the Central Research Fund of the University of London. I would like to thank the CRF for their support. Other computer calculations took place on William Stein’s machine

Mec-cah at Harvard University; I would like to thank him for the use of his

computer.

I would like to thank Kevin Buzzard, Edray Goins and Ken McMurdy for helpful conversations. I would also like to thank the anonymous ref-eree for their extensive comments which improved the exposition and style markedly, and for pointing out an error in an earlier version of the paper.

References

[1] W. Bosma, J. Cannon,C. Playoust, The Magma algebra system I: The user language. J. Symb. Comp. 24(3–4) (1997), 235–265. http://magma.maths.usyd.edu.au.

[2] K. Buzzard, Questions about slopes of modular forms. Astérisque 298 (2005), 1–15. [3] K. Buzzard, F. Calegari, Slopes of overconvergent 2-adic modular forms. Compos. Math.

141(3) (2005), 591–604.

[4] K. Buzzard, L. J. P. Kilford, The 2-adic eigencurve at the boundary of weight space. Compos. Math. 141(3) (2005), 605–619.

[5] H. Cohen, J. Oesterlé, Dimensions des espaces de formes modulaires. Lecture Notes in Mathematics 627 (1977), 69–78.

[6] R. Coleman, Classical and overconvergent modular forms of higher level. J. Théor. Nombres Bordeaux 9(2) (1997), 395–403.

[7] R. Coleman, p-adic Banach spaces and families of modular forms. Invent. Math 127 (1997), 417–479.

[8] R. Coleman, Classical and overconvergent modular forms. Invent. Math. 124(1–3) (1996), 215–241.

[9] M. Emerton, 2-adic Modular Forms of minimal slope. PhD thesis, Harvard University, 1998.

[10] N. Katz, p-adic properties of modular forms and modular curves. Lecture Notes in Mathe-matics 350 (1973), 69–190.

[11] L. J. P. Kilford, Slopes of 2-adic overconvergent modular forms with small level. Math. Res. Lett. 11(5–6) (2004), 723–739.

[12] D. Loeffler, Spectral expansions of overconvergent modular functions. Int. Math. Res. Not. IMRN 2007, no. 16, Art. ID rnm050.

[13] T. Miyake, Modular Forms. Springer, 1989.

[14] K. Ribet, Galois representations attached to eigenforms with Nebentypus. Lecture Notes in Mathematics 601 (1977), 17–51.

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[15] J.-P. Serre, Endomorphismes complètements continus des espaces de Banach p-adique Publ. Math. IHES 12 (1962), 69–85.

[16] L. Smithline, Exploring slopes of p-adic modular forms. PhD thesis, University of California at Berkeley, 2000.

[17] L. Smithline, Compact operators with rational generation. In Number theory, volume 36 of CRM Proc. Lecture Notes, pages 287–294. Amer. Math. Soc., Providence, RI, 2004.

L. J. P Kilford

Department of Mathematics Royal Fort Annexe

University of Bristol BS8 1TW, United Kingdom

E-mail : maljpk@math.bris.ac.uk

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